The Category of Node-and-Choice Forms, with Subcategories for Choice-Sequence Forms and Choice-Set Forms

The literature specifies extensive-form games in several styles, and eventually I hope to formally translate games across those styles. Toward that end, this paper defines NCF, the category of node-and-choice forms. The category's objects are game forms in any style, and the category's isomorphisms are made to accord with the literature's small handful of ad hoc style equivalences. More specifically, the paper develops two full subcategories: CsqF for forms whose nodes are choice-sequences, and CsetF for forms whose nodes are choice-sets. I show that NCF is ``isomorphically enclosed'' in CsqF in the sense that each NCF form is isomorphic to a CsqF form. Similarly, I show that CsqF_\tilde{a} is isomorphically enclosed in CsetF in the sense that each CsqF form with no-absentmindedness is isomorphic to a CsetF form. The converses are found to be almost immediate, and the resulting equivalences unify and simplify two ad hoc style equivalences in Kline and Luckraz (Economic Theory Bulletin, 2016) and Streufert (International Journal of Game Theory, forthcoming). Aside from the larger agenda, this paper makes three practical contributions. Style equivalences are made easier to derive by [1] a natural concept of isomorphic invariance and [2] the composability of isomorphic enclosures. In addition, [3] some new consequences of equivalence are systematically deduced.